# APPENDIX: COPULA FUNCTION AND COPULA DENSITY

## Deriving the Copula Density from Copula Function

Since the copula function *C* is a joint CDF taking as arguments the CDF functions derivatives with respect to x *and y,* implies using the chain rule for derivatives: Taking first the derivative with respect to arguments of *C, F* and *F2,* which are functions of x *and y,* and multiplying each by the derivatives of *F (x), F (y)* with respect to their arguments x *and y.*

The chain rule for derivatives using two functions *g* and *h* of x, such that *k(x) — g[h(x)]* implies that:

In the joint CDF, the copula function *C* is the equivalent of *g* and the joint CDF *F(x)* is the equivalent of *h(x).* Hence with a single function of x, we would write:

The rule applies as well to two functions:

It should be kept in mind that the CDFs *F* are themselves functions of x and *y,* so that the expanded expression of C is:

## Joint Probability Density Functions with Two Variables

The copula function is a joint CDF, equal to the probabilities that the two variables are lower than or equal to x *and y,* respectively. The joint probability density function, or joint PDF, is the probability that two variables following the distributions functions *F (x)* and *F2(x2)* take the joint pair of values x *and y.* probability *(X—x* and *Y — y) —* joint PDF(x, *y)*

Starting from the copula function:

Because it cumulates the probabilities of Xand F being lower than x *and y,* it sums up all joint probabilities of couples of values of *X* and Flower than x *and y.* The copula, being a cumulative function, is the integral of the corresponding joint density function that applies to a couple of values x *and y.* This joint PDF, or joint probability applicable to the pair (x, *y),* is derived from the copula function by taking the derivatives with respect to the two values x and *y,* or the second derivatives with respect to x *and y* respectively:

The derivation of this expression is expanded in the next section.

When the CDF of x is *F(x),* the PDF of x *is fix) — Fx).* Moreover, for two variables, the joint PDF is simply *fix, y).* This makes the notations a bit easier to follow. The two univariate density functions are/^(x) = *F^x)* and *f2(y) = F2(y).* Putting those notations together, we get the joint density function as the product of the two density functions multiplied by the second derivatives of the copula *C* with respect to the CDFs *F* and *F2:*

It is convenient to use as abbreviation for this second derivative of the copula *C,* the small letter c, defined as:

*c(Fv F2)* is the copula density. The copula density provides the joint density function of variables *X*and Fas the copula density multiplied by the marginal (unconditional and univariate) densities (or probabilities) of *X* and *Y, F (x), F (y).* The copula density is a function of the joint density of two variables and their unconditional (marginal) densities.

The copula density is:

This provides the intuitive understanding of the copula density, as the ratio of the joint probability density to the product of the probabilities that *X— x* and *Y — y.* The product is equal to the joint probability of x and *y* occurring only when *X*and *Y* are independent.

• When *X* and Fare independent, the copula density takes the value 1.

• When *X* and Fare positively correlated, the copula density is higher than 1 because the joint probability *fix,y) >fl(x)f2(y).*

• When *X* and Fare inversely related, the copula density is lower than 1 *because fix, y)* < *fx(x)*